Quantitative limit theorems via relative log-concavity

TL;DR

This paper introduces convexity-based tools to establish quantitative limit theorems, providing bounds on distribution discrepancies and applications to various probabilistic approximations.

Contribution

It develops new bounds for total variation discrepancies using relative log-concavity, with applications to geometric, binomial, and Poisson approximations.

Findings

Bounds for total variation discrepancy between measures with log-concavity

Applications to geometric and binomial approximations

Quantitative bounds for infinitely divisible distributions

Abstract

In this paper we develop tools for studying limit theorems by means of convexity. We establish bounds for the discrepancy in total variation between probability measures $\mu$ and $\nu$ such that $\nu$ is log-concave with respect to $\mu$. We discuss a variety of applications, which include geometric and binomial approximations to sums of random variables, and discrepancy between Gamma distributions. As special cases we obtain a law of rare events for intrinsic volumes, quantitative bounds on proximity to geometric for infinitely divisible distributions, as well as binomial and Poisson approximation for matroids.